What Lies Behind Pythagoras’ Theorem

Pythagoras (570-500 BC) was born on the island of Samos in what is now Greece. On the advice of Thales of Miletus, he travelled to Memphis in Egypt where he came into contact with priests renowned for their wisdom. At the age of 40, he fled the tyranny of Polycrates to Croton in Southern Italy, where he established a philosophical and religious community. Those who entered the community’s inner circle were governed by a strict set of ascetic and ethical rules, forsaking personal possessions, assuming a mainly vegetarian diet, and—since words are so often careless and misrepresentative—observing the strictest silence. Some of the community’s more eccentric rules, such as ‘do not break bread’ or ‘do not poke the fire with a sword’ may have been riddles or allegories that required interpreting. Pythagoras’ brotherly community has been hailed as a prototype for later philosophical institutions such as Plato’s Academy, Aristotle’s Lyceum, and Epicurus’ Garden, and, indeed, for the monastic life and associated early universities.

Music played an important role in Pythagoras’ community. Pythagoreans recited poetry, sang hymns to Apollo, and played on the lyre to cure illnesses of both body and soul. It is said that, one day, Pythagoras passed by some blacksmiths at work, and found that their hammering on anvils produced especially harmonious sounds. He then found that the anvils were simple ratios of one another, one being half the size of the first, another two thirds of the size, and so on. This discovery of a relationship between numerical ratios and musical intervals led Pythagoras to believe that the study of mathematics was the key to understanding the structure and order of the universe. According to his ‘harmony of the spheres’, the heavenly bodies move according to mathematical equations that correspond to musical notes and form part of a grand cosmic symphony.

Pythagoras never separated religion from science and philosophy, which, even in his day, left him open to accusations of mysticism. No doubt under the influence of Orphism, an Ancient Greek mystery religion that arose from pre-Hellenic beliefs and the Thracian cult of Zagreus, he believed in the transmigration of the soul; that is, in the reincarnation of the soul over time into the bodies of human beings, animals, or plants (metempsychosis) until such a time as it became moral. He claimed to have lived four lives and to remember them all in great detail, and once recognized the cry of his dead friend in the yelping of a puppy. After his death, the Pythagoreans deified him, and attributed him with a golden thigh and the gift of bilocation (being in two places at once). But in his own lifetime Pythagoras had always been a paragon of modesty, declining to be called a ‘wise man’ or ‘sophos’, and preferring instead to be called ‘a lover of wisdom’ or ‘philosophos’—thereby coining the term ‘philosopher’.

It is said that Numa Pompilius, the second king of Rome, had been schooled by Pythagoras, whence his great wisdom and piety. This story is referred to and discredited by Plutarch and Livy, not least because the dates do not tally, with Pythagoras having lived from about 570 to 500 BC, and so considerably later than the semi-legendary King Numa. Even though Pythagoras and Numa never met, Pythagoras and the Pythagoreans exerted a strong influence on the Roman mind. In the Tusculan Disputations, Cicero indicates that Pythagoras rose to fame in southern Italy at just the same time that Brutus brought an end to the monarchy, and that many Roman usages derived from the Pythagoreans. Unfortunately, he does not elaborate on the nature of these usages. In his Natural History, Pliny the Elder tells us that, in 343 BC, during the war with the Samnites, the god Apollo ordered the Romans to erect one statue to the wisest and another to the bravest of all Greeks, with their choices falling upon Pythagoras for the former and Alcibiades for the latter. Pliny expresses surprise that they picked Pythagoras over Socrates, whom Apollo himself had called the wisest of all men. But the fact is that the Romans liked to think of the Greek-Italian Pythagoras as their very own philosopher, and spun all sorts of stories, such as the one about Numa, to better appropriate him.

Apart from this, Pythagoras also exerted a strong indirect influence on Roman thinking, and indeed on all philosophy and theology, through the teachings of Plato, the principal architect of the western mind. Aristotle, who was Plato’s pupil of twenty years, claimed that his master’s teachings owed much to those of Pythagoras; so much, in fact, that, in his History of Western Philosophy, Bertrand Russell upheld not Plato but Pythagoras as the most influential of all Western philosophers. Pythagoras’ influence is especially evident in Plato’s mystical approach to the soul and in his emphasis on mathematics and, more generally, abstract thinking, as a secure basis for the practice of philosophy.

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Invisibility Cloaks and the Ring of Gyges

George Eleftheriades and Michael Selvanayagam, researchers at the University of Toronto, have designed and tested a new approach to invisibility cloaking. Their method involves surrounding an object with miniature antennae emitting an electromagnetic field that cancels out waves reflecting back from the cloaked object. Although their tests showed the cloaking system to work with radio waves, they see no reason why, as the necessary antenna technology matures, it could not also work with light waves.

All this opens the way for a Harry Potter-style invisibility cloak that is thin, scalable, and adaptable to different types of objects. Some of the uses being touted for this quasi magical cloak include hiding military vehicles and conducting surveillance operations. But what if the cloak falls, as it surely will, into the wrong hands? Have the scientists really thought through the consequences? The infamous banker Bob Diamond spoke of ethics as ‘what you do when nobody’s looking’. If bankers, politicians, and even churchmen can no longer be trusted to do the right thing, then who can? But beyond this, the cloak of invisibility raises important questions about human nature: do intelligent people do the right thing because it is the right thing or because they fear being caught, judged, and punished? More fundamentally, is man innately good, under the direction of his conscience and sense of guilt, or is his restraint rather the product of fear and coercion instilled by a Hobbesian social contract that serves to keep him in check?

In Greek mythology, the Cap of Invisibility or Helm of Darkness is a helmet or cap variously worn by Athena, Hermes, and Perseus to make themselves invisible to gods, heroes, monsters, and men. In Book II of the Republic, Plato discusses the Ring of Gyges, which, according to legend, makes its bearer invisible. The ring was once given to the shepherd Gyges who used it to seduce the Queen of King Candaules and thereby usurp the throne of Lydia. In the Republic, the character of Socrates asserts that justice is the excellence of the soul without which a man cannot live well and be happy, and, therefore, that justice is inherently desirable. However, Glaucon doubts whether to be just is always better than to be unjust. All goods, he says, can be divided into one of three classes: harmless pleasures that are desirable in themselves; goods such as gymnastics, the care of the sick, or the various ways of making money that are desirable for what they bring; and goods such as knowledge, sight, or health that are desirable both in themselves and for what they bring. To which of these three classes does justice belong?

Socrates replies that justice belongs to the third class, but Glaucon points out that most people would disagree and place it firmly in the second class. Indeed, most people think that to do injustice is good, but that to suffer injustice is evil; as the evil outweighs the good, they agree among themselves not to do injustice. If a just man got hold of the Ring of Gyges, he would most certainly behave unjustly, proving that he is just only because he is weak and fears retribution, and not because justice is desirable in itself. The truly just man who cares only for justice and not for the appearance of justice will be thought unjust and suffer every kind of evil until the day he finally understands that he should not be, but only seem, just. In contrast, the unjust man who is resourceful enough to seem just will be thought just and always get the better of everyone and everything. Adeimantus adds that when people praise justice, they praise it for what it brings rather than for itself. Realizing this, the superior man devotes himself not to justice itself but only to its appearance.

Adeimantus claims that he does not truly believe his argument, but is nonetheless pressing it to provoke Socrates into taking its other side and demonstrating that justice is desirable in and of itself. As part of his lengthy reply, Socrates famously conjures up an idealized Republic to help him define justice (or, as he puts it, “locate justice within the State”). After having defined justice in the state and justice in the individual, Socrates asserts that the just man orders his inner life in such a way as to be his own master and his own law. The soul of such a man can be said to be healthy, for justice and injustice are to the soul as health and disease are to the body: virtue is the health and the beauty of the soul, vice its disease and debility. If justice is the health of the soul, and if health is desirable in and for itself, then, by analogy, justice too is desirable in and for itself.

This is as far as Plato gets in the Republic. Notice that his conclusion that justice is intrinsically desirable does not in itself answer the original question, which was whether an intelligent person would still behave justly if he no longer feared being caught and punished. From Plato’s other writings, the answer is surely yes, even if Plato defines ‘intelligent’ in such a way that only he and some of his friends at the Academy actually meet the criteria. These select men are, of course, the famous philosopher-kings.

Book Review: AC Grayling’s Friendship

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It is striking that the great thinkers, from Aristotle to Augustine and Mencius to Montaigne, devoted so much of their time and thought to friendship, but almost none of either to marriage. Grayling’s timely treatise reacquaints us with a great but forgotten good that promises to fulfil so many of our practical, intellectual, emotional, and metaphysical needs. The book principally consists of a history of the philosophy of friendship capped by an account of canonical, often homosexual or homosocial friendships such as that of Achilles and Patroclus and Jonathan and David, who, in the Bible, describes the love of Jonathan as “better even than that of women”. Throughout, Grayling seeks to define friendship and, in so doing, explores its many forms, facets, charms, and consolations.

Perhaps in a desire to be modern, relevant, or politic, Grayling seems to reject the classical notion that, at its best and most meaningful, friendship is a highly elitist good. For the greats, only virtuous men can be ideal friends. Aristotle famously says that, while there are many ways for men to be bad, there is only one way for them to be good, and it is precisely in this sense that an ideal friend is ‘another self’—a historically important notion that Grayling severally dismisses. Because they are all one and the same, virtuous men are predictable, reliable, and therefore worthy of one another’s friendship. In contrast, bad people are in some way unlike themselves, and just as likely to hate other bad people as anyone else.

In my opinion, Plato, whom Grayling underrates, advances by far the most subtle and sophisticated of all theories of friendship, one far superior even to that of Aristotle. Despite the extravagant praise that he lavishes upon friendship, Aristotle is quite clear that the best and happiest life is not that spent in friendship, but in the contemplation of those things that are most true and therefore most beautiful and most dependable. There is a contradiction here: if the best life is a life of contemplation, then friendship is either superfluous or inimical to the best life, and therefore undeserving of the high praise that Aristotle lavishes upon it. It may be, as Aristotle tentatively suggests, that friendship is needed because it promotes contemplation, or that contemplation is only possible some of the time and friendship is needed the rest of the time, or even that a life of friendship is just as good as a life of contemplation. So much for Aristotle, one might say.

Plato’s Lysis may seem to fail in its task of defining friendship, but one should never take Plato or his mouthpiece Socrates at face value. There is far more to the Lysis than a couple of interesting but misguided thoughts on friendship. By discussing friendship with Lysis and Menexenus as he does, Socrates is not only discussing friendship, but also demonstrating to the youths that, even though they count each other as close friends, they do not really know what friendship is, and that, whatever friendship is, it is something far deeper and far more meaningful than the puerile ‘friendship’ that they share. In contrast to the youths, Socrates knows perfectly well what friendship is, and is only feigning ignorance so as to teach the youths: ‘…and I, an old boy, who would fain be one of you…’ More than that, by discussing friendship with Lysis and Menexenus as he does, Socrates is himself in the process of befriending them. He befriends them not with pleasant banter or gossipy chitchat, as most people ‘befriend’ one another, but with the kind of philosophical conversation that is the hallmark of the deepest and most meaningful of friendships. In the course of this philosophical conversation, he tells the youths that he should ‘greatly prefer a real friend to all the gold of Darius’, thereby signifying not only that he places friendship on the same high pedestal as philosophy, to which he has devoted (and will sacrifice) his life, but also that the kind of friendship that he has in mind is so rare and uncommon that even he does not possess it. If friendship ultimately escapes definition, then this is because, like philosophy, friendship is not so much a thing-in-itself as it is a process for becoming. True friends seek together to live truer, fuller lives by relating to each other authentically and by teaching each other about the limitations of their beliefs and the defects in their character, which are a far greater source of error than mere rational confusion. For Plato, friendship and philosophy are aspects of one and the same impulse, one and the same love: the love that seeks to know.

Just as philosophy leads to friendship, so friendship leads to philosophy. In the Phaedrus, Plato’s most important work on friendship (although not generally recognized as such—Grayling fails to mention it), Socrates and Phaedrus go out into the idyllic countryside just outside Athens and have a long conversation about the anatomy of the soul, the nature of true love, the art of persuasion, and the merits of the spoken over the written word. At the end of this conversation, Socrates offers a prayer to the local deities. This is the famous Socratic Prayer, which is notable both in itself and for the response that it elicits from Phaedrus.

Socrates: Beloved Pan, and all ye other gods who haunt this place, give me beauty in the inward soul; and may the outward and inward man be at one. May I reckon the wise to be the wealthy, and may I have such a quantity of gold as a temperate man and he only can bear and carry. —Anything more? The prayer, I think, is enough for me.

Phaedrus: Ask the same for me, for friends should have all things in common.

Plato may fail to define friendship in the Lysis, but in the Phaedrus he gives us its living embodiment. Socrates and Phaedrus spend their time together enjoying the beautiful Attic countryside while engaging in honest and open philosophical conversation. By exercising and building upon reason, they are not only furthering each other’s understanding, but also transforming a life of friendship into a life of joint contemplation of those things that are most true and hence most beautiful and most dependable. If only on the basis of his response to the Socratic Prayer, it is obvious that Phaedrus is another self to Socrates, since he makes the same choices as Socrates and even justifies making those choices on the grounds that their friendship requires it. Whereas Aristotle and Grayling try to tell us what friendship is, Plato lets us feel it in all its allure and transformative power.

Spectator Review of ‘Plato: Letters to my Son’

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Plato: Letters to My Son : Video

My New Book

My new book (and first novel) Plato, Letters to my Son is now available for pre-order on amazon.co.uk.

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God and the Meaning of Life

Can a belief in God give our life its meaning?

Historically and still today many people feel that humankind was created by a supernatural entity called God, that God had an intelligent purpose in creating humankind, and that this intelligent purpose is the ‘meaning of life’.

Here is not the place to go through the various arguments for and against the existence of God. Suffice to say that many people who believe in God would admit that they do not really know what God’s purpose might be, nor that it would necessarily be particularly meaningful. For example, the second Law of Thermodynamics states that entropy increases up to the point at which equilibrium is reached, and God’s purpose in creating us and, indeed, all of nature, might simply have been to catalyse this process. If our God-given purpose is to act as super-efficient heat dissipaters, then this purpose is almost as bad as no purpose at all.

In fact, one might argue that having no God-given or pre-determined purpose is better than having any sort of pre-determined purpose at all (even a more traditional and uplifting one such as serving the will of God or improving our karma) because it frees us to be the authors of our own purpose or purposes, and so to lead truly dignified and meaningful lives. Thus, even if God exists, and even if God had an intelligent purpose in creating humankind, we do not know what this purpose is and, whatever it is, we would much rather be free to determine our own purpose or purposes.

Some might object that not to have a pre-determined purpose is, really, not to have any purpose at all. However, this is to believe (1) that for something to have a purpose, it must have been created with that purpose in mind, and (2) that something that was created with a purpose in mind must necessarily have that very purpose for which it was created. Last summer, I visited Château-Neuf-du-Pape in the Southern Rhone where I picked up a beautiful rounded stone called a galet from one of the vineyards, took it back to England, and put it to excellent use as a book-end. The purpose of these stones in the vineyard is to absorb the heat from the sun during the daytime and then to release it during the night time. However, galets were not created with this or any other purpose in mind. Even if galets were created with a purpose in mind, then this purpose was almost certainly not (1) to make great wine, (2) to serve as book-ends, or (3) to be beautiful. That same evening over some supper, I had my wine-loving friends to blind-taste a bottle of claret that I had brought along from England. Unfortunately, I did not have a decanter to hand, so I masked the identity of the wine by slipping the bottle into one of my (clean) dark blue socks. Unlike the galet, the sock had been created with a purpose in mind, even if this purpose was a very different one from the one that it eventually found.

Some might also or otherwise object that talk about the purpose of life is neither here nor there because life is merely a prelude to some form of eternal afterlife and this is, if you like, its purpose. But (1) it is not at all clear that there is or even can be some form of eternal afterlife that involves the survival of the personal ego. (2) Even if there is an eternal afterlife, living for ever is not a meaning in itself and so the question arises, what is the meaning of the eternal afterlife? If the eternal afterlife has a predetermined purpose, again, we do not know what this purpose is and, whatever it is, we would much rather be free to determine our own purpose or purposes, which we can just as well do in this life. (3) It is not just that reliance on an eternal afterlife merely postpones the question of life’s purpose, but also that it prevents us from determining a purpose or purposes for what may well be the only life that we do have. (4) If one believes that it is the brevity or finiteness of human life that lends it shape or meaning, then an eternal afterlife cannot, by definition, have any purpose. I do not personally believe that the brevity or finiteness of human life lends it shape or meaning, and rather suspect that this is just another ego defense against death. However, that is quite another debate and I shall put it to one side.

The real point here is that whether or not God exists, whether or not God has a purpose for us, and whether or not there is an afterlife, we should strive to give meaning to our lives. For unless we can be free to determine our own purpose or purposes, our life may, at worse, have no purpose at all, and, at best, only some unfathomable pre-determined purpose that is not of our choosing. The great philosopher Plato once defined a human being as an animal, biped, featherless, and with broad nails, but a much better definition that he gave was simply this, ‘A being in search of meaning.’

Adapted from The Art of Failure: The Anti Self-Help Guide

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